Table Of ContentStrong and complex electron-lattice correlation in optimally doped Bi Sr CaCu O
2 2 2 8+δ
G.-H. Gweon,1,2 S. Y. Zhou,1 M. C. Watson,1 T. Sasagawa,3,4 H. Takagi,3,4,5 and A. Lanzara1,6
1Department of Physics, University of California, Berkeley, CA, 94720
2Department of Physics, University of California, Santa Cruz, CA, 95064
3Department of Advanced Materials Science, University of Tokyo, Kashiwa, Chiba 277-8561, Japan
7
4CREST, Japan Science and Technology Agency, Saitama 332-0012, Japan
0
0 5RIKEN (The Institute of Physical and Chemical Research), Wako 351-0198, Japan
2 6Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA, 94720
(Dated: February 6, 2008)
n
a We discuss the nature of electron-lattice interaction in optimally doped Bi2Sr2CaCu2O8+δ sam-
J ples, using isotope effect (IE) in angle resolved photoemission spectroscopy (ARPES) data. The
4 IE in the ARPES line width and the IE in the ARPES dispersion are both quite large, implying a
1 strongelectron-latticecorrelation. Thestrengthoftheelectron-latticeinteractionis“intermediate,”
i.e.strongerthantheMigdal-Eliashbergregimebutweakerthanthesmallpolaronregime,requiring
] a more general picture of the ARPES “kink” than the commonly used Migdal-Eliashberg picture.
n
The two IEs also imply a complex interaction, due to their strong momentum dependence and
o
theirdifferingsign behaviors. Insum,weproposean intermediate-strengthcouplingof electronsto
c
- localized lattice vibrations via charge density fluctuations.
r
p
PACSnumbers: 74.25.Jb,74.72.-h,79.60.-i,71.38.-k
u
s
.
t In the last few years, several angle resolved photoe- unique method that can directly sort out the role of the
a
m mission spectroscopy (ARPES) reports suggested a sig- lattice to shed light on this complexity.
nificant interaction of dynamic lattice distortions with In this Letter, we provide evidence of an anomalously
-
d electrons in the cuprate superconductors [1, 2, 3, 4], but large isotope effect (IE) in the ARPES peak width for
n the basic nature of the electron-lattice interactionis still optimally doped Bi Sr CaCu O superconductor, in
2 2 2 8+δ
o
unclear. addition to the IE in the ARPES peak position reported
c
[ Ononehand,theMigdal-Eliashbergtheory,astandard earlier [3]. Detailed comparison of the data with vari-
modelofthe electron-phononinteractioninsolids,seems ous theories of the electron-lattice interaction provides
1
to provide a basic framework for explaining [5, 6] a key compelling evidence that the strength of the interaction
v
1 featureofARPESdata,i.e.the“kink”intheARPESdis- is intermediate, i.e. beyond the Migal-Eliashberg regime
9 persion[1,2,4]. Inthistheory,theARPESkinkappears but not in the small polaron regime. This and another
2 at phonon energy, ω . For excitation energy (ω) smaller feature,thelocalnatureofthelatticevibrationsinvolving
p
1 than ω , long-lived quasi-particles (electron dressed by only a few lattice sites [7, 21], are an important refine-
p
0
virtual phonons) form. For ω >ω , short-livedelectrons ment of an anisotropic electron-lattice interactionmodel
7 p
0 scatter strongly with real phonons. For ω ≫ ωp, the ef- proposed before [2, 4, 6]. Finally, we believe that our
/ fect of the scattering becomes negligible. This familiar resultsgiveimportantcluesforunderstandingthenodal-
t
a Migdal-Eliashberg picture has been also generalized to anti-nodaldichotomyobservedinthecuprates[18,22]as
m
cases where other bosonic excitations are involved[7, 8]. well as the interplay between the electron-electroninter-
- On the other hand, a large number of experimental action and the electron-lattice interaction, as proposed
d
reports[3, 9,10, 11,12, 13,14,15, 16,17,18]suggestan recently [3, 23, 24].
n
o interactionstrengthbeyondtheMigal-Eliashbergtheory. We discuss data taken at low temperature (25 K),
c Such a proposal is plausible also from the ARPES point where the electron-lattice interaction is strongly en-
: of view, because as momentum approaches the Brillouin hanced[3]wellbelowT∗ (pseudo-gaptemperature)≈T
v c
i zone boundary, ωp (≈ 70 meV) becomes comparable to (92 K for 16O and 91 K for 18O). Details of the experi-
X
the effective bandwidth governedby Fermivelocity[19], ment are described elsewhere [3]. Throughout this Let-
r weakening the Migal-Eliashberg theory, and the actual ter, we use the term “high energy” or “low energy” to
a
quasi-particle weight Z measured by ARPES is rather refer to the magnitude |E −E | (E is the Fermi en-
F F
small, ∼ 0.1 [20]. ergy),highorlow meaning relativeto the kink energyin
To date, there has been no ARPES study reconcil- particular.
ing these two important points of view. This absence In Figure 1 we show raw ARPES data along two im-
appears to be due to the complexity of the cuprate portant momentum space cuts, cut a (nodal) and cut b
physics, which makes it difficult to disentangle the effect (off-nodal or near-anti-nodal [25]). A more detailed mo-
of electron-lattice interaction from the effect of strong mentumdependenceisdescribedelsewhere[3,26]forthe
electron-electron interaction. As we demonstrated re- peak position and below for the width [25]. The energy
cently [3], ARPES on isotope substituted samples is a distributioncurves(EDCs;intensityasfunctionofenergy
2
(a) (b) M Y realistic [28] choice of a fewer number of high frequency
1168OO αba modes, while doing a better job of describing a smooth
β kink [3] seen in the data. We use a gap-less linear band
16O
R Γ with v = 1.8 eV˚A for ǫ(k), and 0.5 for the dimension-
) F
u. less electron-phonon coupling constant λ [5]. For 18O,
a. α α only ω is modified, in accordance with the harmonic
( p
y phonon model. We have checked extensively that use of
t
si additional features such as anisotropic coupling [29], full
n
e band structure and superconducting gap [6, 7, 8] does
t
n
I β ME not affect the discussion presented below.
In Figure 2, we present a detailed analysis of the IE
HWHMR SS SS β introduced in Figure 1. Panels a and b show “peak IE,”
namely the isotope dependence of EDC peak position.
-0.2 0 -0.2 0 Likewise,panelscanddshow“widthIE,”theisotopede-
E – E (eV) pendence of EDC peak width. Several important points
F
can be made. First, both IEs are large at high energy,
FIG. 1: Comparison of ARPES EDCs for three samples while small at low energy [30]. This result shows a com-
with different isotope treatments, normalized to same peak plete agreement with the previous work [3] as the peak
height; normal (16O), isotope-substituted (18O) and isotope
IE is concerned, and further shows that the kink energy
re-substituted (16OR) samples for cuts a and b, as indicated
(≈thearrowpositioninpanelbforbothcuts)continues
intheinsetofpanelbalongwiththeFermisurface. Datafor
◦ toseparatesmallIEandlargeIEalsoforthewidthIE,as
each angle were integrated over a 1 angle window in favor
of statistics. The inset saddling the two panels shows Migal- it separatesthe sharpcoherentpeak and the broadinco-
Eliashberg simulations for 16O and18O,as described in text. herentpeak[3]. Second,theMigal-Eliashbergsimulation
(light red lines in b,d) poorly explains the data in gen-
eralandthedataathighenergyinparticular. Third,the
at fixed momentum value) displayed here show clear IE IEs are enhanced for the off-nodal cut, consistent with a
for both the peak position and the peak width. Previ- strongly momentum dependent electron-lattice coupling
ously [3], momentum distribution curves (MDCs; inten- [6] and making the failure of the Migal-Eliashberg the-
sityasfunctionofmomentumatfixedenergyvalue)were ory less severe for the nodal cut. Fourth, the sign of
used to discuss IE in the MDC dispersion and a small the width IE is unchanged going from cut a to cut b,
red-shiftofthe kink energy. In this Letter,we gofurther i.e. the peak always sharpens up for 18O, while the sign
to discuss the line width, which requires more care in of the peak IE changes from cut a to cut b. These dif-
analysis. For this reason, we discuss EDCs rather than
MDCs and use the half width at half maximum on the
right side (HWHM ) of EDC peak to quantify the peak (a) (b)
R peak IE peak IE
width[27],asillustratedinpanela. Aswewillshow,the
EDC analysisgives the same resultas the MDC analysis 20 cut a cut b (x-1)
in terms of the peak position.
)
V
Two observations can be added [3] to support the in- e ME
trinsic nature of the observed IE. First, high energy m0
(
peaks in Figure 1 sharpen up upon isotope substitution, y (c) (d)
g width IE width IE
reinforcingtheargument[3]thatasimplebandstructure r
e
effect or a simple disorder effect, arising from a stronger En20 cut b
disorder for the 18O sample due to the substitution pro- cut a
cess,cannotbetheoriginoftheobservedIE. Second,the
ME
EDCs for the re-substituted sample (green) are nearly
0
identical to those of the 16O sample.
-0.2 -0.1 0 -0.2 -0.1 0
It is quite clear that the Migal-Eliashberg theory (in-
E – E (eV)
set) predicts a small IE overall,in strong contrastto the F
data. For the Migal-Eliashberg theory, we chose the fi-
FIG. 2: (a,b) IE in the EDC peak positions for cuts a and
nite temperature formalism [5] for electron coupled to
b. (c,d)IEintheEDCwidths(HWHMR). TheIEisdefined
sevenEinstein phononmodes, equally spacedfrom 10 to as 16O−18O, and note that thesign-reversed IE is plotted in
70meV,withthemaximumdeterminedbyneutronscat- panel b. The horizontal energy axis correponds to the peak
tering data [28]. For this model, we refer to this max- position for the 16O sample. Gray dashed lines are IEs for
imum frequency as ωp. For the purpose of this paper, 16OR. For comparison Migal-Eliashberg simulation results
this choice of phonon frequencies is equivalent to a more are shown as light red lines in panels b,d.
3
V) 20 high energy M Y tfoiorn18sOtr.enSgetchongdo,inmgobdeeylosn[d32t,he33M] iignaclo-rEploiarsahtbinegrginttheeroarcy-
e
m ∆ were shown to be able to explain the large (small) iso-
( tope effect at high (low) energy and also the sharpening
E
I of ARPESpeak induced by 18O. Onthe other hand, the
h
dt 0 sign change of Figures 2a,b could not be addressed, pre-
Wi low energy sumably due to the simple form of interaction assumed.
Γ
It is significant to note that in these calculations the
0 10 20
∆ (meV) binding energy of peak increases while the peak width
decreases, a non-trivial result in qualitative agreement
FIG. 3: IE of the EDC width as a function of the super- with our nodal cut data (Figures 2a,c).
conducting gap ∆, for six momentum cuts [25] shown in the Basedonthese results,we proposethatacombination
right panel. For each cut, low energy (-70 meV to 0 meV)
ofthese two approachesis requiredto be totally success-
average and high energy (-250 meV to -70 meV) average of
ful, as we further elaborate now.
IEareshownwitherrorbars. Thelineshownforhighenergy
IE is guide tothe eye. InFigures4a,b,weshowlineshapesimulationsforcut
busingtheHolsteinmodelinthestrong-coupling(orthe
anti-adiabatic) limit where a simple perturbative solu-
tion can be obtained [34]. For input parameters of the
fering sign behaviors are difficult to understand within
Holstein model, we use ω = 70 meV (66 meV for 18O)
the usual self energy analysis scheme [31], because the p
andtightbindingfitofǫ(k)[35]tothe16OARPESdata.
Kramers-Kronig relationship connects the real part and
The interactionparameter λ was taken to be isotope de-
theimaginarypartofselfenergy[31],determinedmainly
pendent [7], 0.9 for 16O and 0.8 for 18O, in order to give
bythe peak positionandwidthrespectively. This means
the correct sign for the peak shift. Note that the peak
that the self energy analysis scheme as it has been em-
sharpening for 18O remains true even if λ is taken to be
ployed to date under two key assumptions, momentum
isotope independent. The essential characteristics of the
independent self energy and high energy data cutoff at
≈0.3−0.4eV,isunreliabletotheaccuracyrelevanthere. line shape simulationis the strongmulti-phonon“shake-
up” features, which occur at harmonics of ω (panel a)
p
InFigure3,wesummarizethemomentumdependence
andwhichwepostulatewillbroaden,duetophononcon-
of the width IE up to cut b [25]. The low energy IE
tinuumandstrongelectron-electroninteractioninsolids,
is small, and vanishes within error bars. In contrast,
into a single peak (panel b), as was also suggested re-
the high energy IE is finite and shows an approximately
cently [18]. In this way, the simulation in panel b suc-
linearcorrelationwiththesuperconductinggapuptocut
cessfullyreproducesthesmallIEforsharplow-energyco-
b [25], as the peak IE [3]. The corresponding IEs for the
Migal-Eliashbergtheoryisnegligible,2meVatthemost.
traTchede bfaacilkurteotohfetthhee sMinigglael-pEhloiansohnbleorogptahpeporroyxicmaantiobne nits) (a) 1168OO ωp EF(c) ωp CP
for electronself energy,subsequently resulting in a small u Multi- CP
IE (shown by light red lines in Figures 2b and 2d) gov- . Phonon CP y
b Peaks g Multi-Phonon
ernedatthemostbytheisotopedependenceofthesingle ar er Peaks
phononenergy. Forinstance, withinthis theorythe high ( (b) n
y e IP
energy line width is given by ≈ωp, explaining the small sit IP
IE at high energy (Figure 2d) as well as explaining why en CP ε(k)
the ARPES width of this theory is by a factor of ≈ 2 t
n
too small in comparison to the observed width (Figure I -0.6 0 k
F
1). In the literature, there have been two different ap- E – E (eV) momentum
F
proaches to remedy this failure of the Migal-Eliashberg
theory. First, Seibold and Grilli [7] used a more general FIG. 4: (a,b) Simulation of EDCs at k=kF for cut b using
Migal-Eliashbergmodelwhichinvolvesλthatisstrongly a small polaron theory [34]. The line shape simulations in-
momentum and isotope dependent. The physicalreason, clude a 10 meV (panel a) and an additional 100 meV (panel
especiallyforthelatter,isthatthebosonthatcouplesto b) FWHM Lorentzian broadening, the latter only for multi-
phonon peaks. (c) Proposed schematic diagram of ARPES
electrons in this case is critical charge order fluctuations
kink for intermediate and weak electron-lattice interaction.
of correlation length of a few lattice constants, instead
ThethicknessofthelineusedforCPorIPdispersioncrudely
of phonons extended throughout the lattice. Using this
represents theARPES line width. The intensity of CP or IP
generalization,it was shown that the model can qualita- is not indicated, but may be roughly inferred from theinset,
tively explain the sign change seen in Figures 2a,b but whichcorrespondstothemainpanelwhatpanelaistopanel
nottheIEseeninFigures2c,d,predictingabroaderpeak b. Seetext for more discussion.
4
herentpeak(CP)andthe largeIEforbroadhigh-energy Materials Sciences and Engineering of the U.S Depart-
incoherent peak (IP). As shown, IEs as large as ≈ 30 mentofEnergyunderContractNo.DEAC03-76SF00098
meV are generic features of this simulation. In addition, and by the National Science Foundation through Grant
the magnitude of the ARPES line width is also realistic. No. DMR-0349361.
These important results make an explicit demonstration
of the multi-phonon physics proposed within a dynamic
spin-Peierls picture [3].
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[23] A.Seidel,H.-H.Lin,D.-H.Lee,Phys.Rev.B71,220501
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not identical, to ǫ(k) [32, 37]. Within this picture, the (2005).
[24] H. C. Fu, C. Honerkamp, and D.-H. Lee, Euro-
kinkanomalystilloccursatωp,theonsetofmulti-phonon phys.Lett. 75, 146 (2006).
continuum (gray area), consistent with a small isotope- [25] Themomentumdependenceoftheisotope effectbeyond
induced shift [3]. Note that this picture places EDCs the off-nodal cut b is underinvestigation.
as more basic quantities than MDCs. Subsequently, the [26] The 16O data in Figure 1b were shifted by 4 meV in
kink energy is best defined as the separation energy be- energy, in order to offset the superconducting gap dif-
ference, with negligible consequence on the main repro-
tween CP and IP branches of EDCs, and the MDC dis-
ducible IE at high energy [3].
persion with a kinky crossover (dotted orange line) is
[27] We use EDC HWHMR, because the large momentum-
merelyaconsequenceofweighttransferbetweenthe two
asymmetric background intensity at high energy makes
branches. unreliableboththeMDCwidthextractionandtheEDC
To conclude, we have shown that IEs in ARPES peak left-width.
width and peak position give important clues to the na- [28] R.J.McQueeney,etal.,Phys.Rev.Lett.82,628(1999).
ture of electron-lattice interaction in optimally doped [29] T. P. Devereaux,privatecommunication.
[30] The temperature dependence (not shown) of the IE in
Bi Sr CaCu O . Theoretical [24] and experimental
2 2 2 8+δ
theARPESwidth iscompletely similar tothatreported
[21]worksfurthersupportthe emergingpicture inwhich
for theARPES dispersion [3].
a cooperationof local lattice distortions and charge/spin [31] A. A.Kordyuk,et al., Phys.Rev.B 71, 214513 (2005).
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This work was supported by the Director, Office of [33] S.FratiniandS.Ciuchi,Phys.Rev.B72,235107(2005).
Science, Office of Basic Energy Sciences, Division of [34] A. S. Alexandrov and J. Ranninger, Phys. Rev. B 45,
5
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strongcouplingperturbationtheoryvalid,withlittlecon- 036402 (2004).
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